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We looked at two questions:
 Find a route that travels on every road exactly once
 (Sometimes called The Gritter Problem)
 Find a route that visits every city exactly once
 (The Travelling Salesman Problem)
We claim that the first is easy, and the second is hard.
The Gritter Problem is also known as "Finding
an Eulerian Cycle." 

In particular, the gritter problem can be solved in Polynomial Time,
time that only grows as a polynomial function of the size of the
instance. Such a problem is said to be in class P, which is
the class of all Polynomial Time problems.
The second problem appears not to be in P.
For more details and a less gentle discussion of
the Time Complexity of the Travelling Salesman Problem,
the wikipedia article is a reasonable reference:
Follow the link to "Computational Complexity."


Specifically,
for every algorithm we currently have for solving the
Travelling Salesman Problem,
the time taken (as a function of the size of the instance) grows
faster than any polynomial function. At the time of writing the
best known algorithms take time proportional to which
grows pretty quickly (although not as quickly as the obvious
algorithm, which grows in time proportional to n!. )
Fair enough. The Gritter Problem is easy, the Travelling Salesman Problem
is hard. That's life.
But there's a sting in the tail. While finding a solution to the second problem seems to be hard, checking an alleged solution is really easy. For a given instance, if you claim that you have found a way to do it, checking that you're right is quick and simple.
In technical terms, checking your alleged
solution only takes Polynomial Time.


I follow your route, mark the cities I visit, then check I got them all.
And if you make your map 100 times bigger, it still only takes 100 times
longer to check if you're right.
So this is a little puzzling.
We have a problem where for each question it's really easy to
check an alleged solution, but (as yet) no way even to know if
such a solution exists.
This leads to an obvious solution strategy.
 Guess a solution
 Check it
 If it's not right, go back to the start
 Print solution and terminate


The "guessing" part means this is "Nondeterministic" and the
checking part takes polynomial time, so this is called a
"Nondeterministic Polynomial" (NP) solution attempt, and any
problem that lends itself to this sort of "solution" method
is called an "NP" problem.
The problem with this "solution" is that for some problems it's
very unlikely ever to terminate, so while it might be that it
never gives you a wrong answer, it might fail to give you an
answer at all.
But all is not lost. Even though we might not be able to solve
NP problems efficiently, we now have a handle on the question
of how hard problems might be.
The next section gives us our first real result.
Prev: What is P? / Next: P Problems Are In NP.
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